Feller–Tornier constant

In mathematics, the Feller–Tornier constant C_FT is the density of the set of all positive integers that have an even number of distinct prime factors raised to a power larger than one (ignoring any prime factors which appear only to the first power). It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982)

${\begin{aligned}C_{\text{FT}}&={1 \over 2}+\left({1 \over 2}\prod _{n=1}^{\infty }\left(1-{2 \over p_{n}^{2}}\right)\right)\\[4pt]&={{1} \over {2}}\left(1+\prod _{n=1}^{\infty }\left(1-{{2} \over {p_{n}^{2}}}\right)\right)\\[4pt]&={1 \over 2}\left(1+{{1} \over {\zeta (2)}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)\right)\\[4pt]&={1 \over 2}+{{3} \over {\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{{1} \over {p_{n}^{2}-1}}\right)=0.66131704946\ldots \end{aligned}}$

(sequence A065493 in the OEIS)

Omega function

The Omega function is given by

$\Omega (x)={\text{the number of prime factors of }}x{\text{ counted by multiplicities}}$

The Iverson bracket is

$[P]={\begin{cases}1&{\text{if }}P{\text{ is true,}}\\0&{\text{if }}P{\text{ is false.}}\end{cases}}$

With these notations, we have

$C_{\text{FT}}=\lim _{n\to \infty }{\frac {\sum _{k=1}^{n}[\Omega (k){\bmod {2}}=0]}{n}}={1 \over 2}$

Prime zeta function

The prime zeta function P is give by

$P(s)=\sum _{p{\text{ is prime}}}{\frac {1}{p^{s}}}.$

The Feller–Tornier constant satisfies

$C_{\text{FT}}={1 \over 2}\left(1+\exp \left(-\sum _{n=1}^{\infty }{2^{n}P(2n) \over n}\right)\right).$

See also

• Riemann zeta function
• L-function
• Euler product
• Twin prime